Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

Abstract

Let us fix a prime p and a homogeneous system of m linear equations $$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ a j , 1 x 1 + ⋯ + a j , k x k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m with coefficients $$a_{j,i}\in \mathbb {F}_p$$ a j , i ∈ F p . Suppose that $$k\ge 3m$$ k ≥ 3 m , that $$a_{j,1}+\dots +a_{j,k}=0$$ a j , 1 + ⋯ + a j , k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m and that every $$m\times m$$ m × m minor of the $$m\times k$$ m × k matrix $$(a_{j,i})_{j,i}$$ ( a j , i ) j , i is non-singular. Then we prove that for any (large) n, any subset $$A\subseteq \mathbb {F}_p^n$$ A ⊆ F p n of size $$|A|> C\cdot \Gamma ^n$$ | A | > C · Γ n contains a solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to the given system of equations such that the vectors $$x_1,\dots ,x_k\in A$$ x 1 , ⋯ , x k ∈ A are all distinct. Here, C and $$\Gamma $$ Γ are constants only depending on p, m and k such that $$\Gamma <p$$ Γ < p . The crucial point here is the condition for the vectors $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k in the solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to be distinct. If we relax this condition and only demand that $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

Partition Rank and Partition Lattices

“…A straightforward application of slice rank was used by Tao [29] to show that there is a constant 1 ≤ Γ < p so that any subset of F n p with size at least Γ n contains a "non-trivial" solution to a given balanced system. This was extended by Sauermann [25] who proved an analogous result guaranteeing solutions where x 1 , . .…”

“…So, for any A ⊆ F n p , A k contains a solution where the x i 's take on just one value. We deem these solutions as "trivial" and all other solutions as "non-trivial", which is consistent with terminology in [25]. So, we are concerned with lower bounds on |A| that force a "non-trivial" solution of varying type.…”

“…The work of Ge and Shangguan [15] uses the slice rank method together with a non-diagonal 3-tensor to find polynomial upper bounds on sizes of subsets in F n q (q a power of an odd prime) that avoid a right angle, drastically improving the previous exponential bound due to Bennett [3]. The work of Sauermann [25] uses a non-diagonal tensor together with a characterization of slice rank by Sawin and Tao [30] to find exponentially small lower bounds on sizes of sets in F n p that guarantee the existence of solutions to certain linear systems where variables are all distinct. However, non-diagonal tensors become more difficult to work with when applying the partition rank method.…”

Intersective sets over abelian groups

“…However, when k ≥ 2, it seems challenging to construct such a diagonal tensor. Recently, Sauermann [22] found a way to bound the slice rank of the nondiagonal tensors, which might be helpful to obtain reasonable bounds for this generalized problem. We also hope to seek more efficient tools to study such problems in the future.…”

“…(2) Let G = Z n and let k be a divisor of n such that 5,6,7,8,9,12,13,14,15,16,17,20,22,24,26,28,31,32,33,34,35,36,39,40,41,42,43,46,47…”